Exact calculation: fractions, surds and π worksheet — GCSE Higher

1.A pizza is cut into 12 equal slices. Ben eats 5 slices and Mia eats 3 slices. What fraction of the pizza is left, giving your answer in its simplest form?

2.Expand and simplify √3(2 + √12).

3.Expand and simplify (2 + √3)², giving your answer in the form a + b√3.

4.A shape is made by joining a rectangle measuring 10 cm by 6 cm to a semicircle of diameter 6 cm along one of the rectangle's shorter sides. Work out the exact area of the shape, in terms of π.

5.Simplify √8 + √18, giving your answer in the form k√2.

6.Rationalise the denominator of 10/(4 − √6), giving your answer in its simplest form.

7.A cake recipe needs 3/4 of a kilogram of sugar. Aisha wants to make half the recipe. Work out how much sugar she needs, giving your answer as a fraction of a kilogram in its simplest form.

8.Simplify √45.

9.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.

10.A quarter-circle has a radius of 6 cm. Work out the exact perimeter of the quarter-circle, giving your answer in terms of π.

11.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.

12.Work out 5/6 × 2/9, giving your answer in its simplest form.

13.Work out √3 × √12, giving your answer as an integer.

14.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.

15.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.